Yesterday, my departmental cross partners and I discussed a technical solution. Yesterday, my departmental partners and I discussed a technical solution.

“Cross partners” is unclear in English. If you mean colleagues from another department, you could say “cross-departmental partners”. Otherwise, just “departmental partners” is simpler and natural.

Yesterday, my departmental crosscross-functional partners and I discussed a technical solution. Yesterday, my cross-functional partners and I discussed a technical solution.

"Cross-functional" is the adjective for people working on different teams, at least in the United States (I *think* it works in Australian and British)

During the meeting, I lost my temper with one of my partners, because he only thought about his own system, and kept proposmaking unreasonable demands. During the meeting, I lost my temper with one of my partners, because he only thought about his own system, and kept making unreasonable demands.

We usually say “make demands” rather than “propose demands.”

I yelled at him, and told him that he should think about the big picture instead of just thinking about himself. I yelled at him and told him that he should think about the big picture instead of just thinking about himself.

No comma needed before “and” when connecting two verbs with the same subject.

But after the meeting, I thought back on itreflected back on my behavior and felt a little regretful. But after the meeting, I reflected back on my behavior and felt a little regretful.

“Reflected on my behavior” is more precise and natural than “thought back on it.”

I will remember this experience and handle things more elegantly instead of roughly in the future. I will remember this experience and handle things more elegantly in the future.

Roughly technically makes sense, but feel a bit unnatural (in US English). We don't generally talk about people "behaving roughly" - although we *do* talk about people behaving "elegantly." I also don't know why, but "more elegantly" feels better to me.